Cocompact Fuchsian groups with a modular embedding

A Fuchsian group $Γ$ has a modular embedding if its adjoint trace field is a totally real number field and every unbounded Galois conjugate $Γ^σ$ comes equipped with a holomorphic (or conjugate holomorphic) map ${ϕ^σ: \mathbb{B}^1 \to \mathbb{B}^1}$ intertwining the actions of $Γ$ and $Γ^σ$ on the Poincaré disk $\mathbb{B}^1$. This paper provides the first cocompact nonarithmetic Fuchsian groups with a modular embedding that are not commensurable with a triangle group. The main result, proved using period domains, is that any immersed totally geodesic complex curve on a complex hyperbolic $2$-orbifold has a modular embedding. Another consequence is arithmeticity of totally geodesic curves on finite-volume complex hyperbolic surfaces that are commensurable with quotients of $\mathbb{B}^1$ by the group generated by reflections in quadrilaterals satisfying certain angle conditions.